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Difference between revisions of "1995 AHSME Problems"

(→‎Problem 20)
(It will all be in wiki form in a few minutes...)
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== Problem 1 ==
 
== Problem 1 ==
 +
Kim earned scores of 87,83, and 88 on her first three mathematics examinations. If Kim receives a score of 90 on the fourth exam, then her average will
 +
 +
A. remain the same
 +
B. increase by 1
 +
C. increase by 2
 +
D. increase by 3
 +
E. increase by 4
  
 
[[1995 AMC 12 Problems/Problem 1|Solution]]
 
[[1995 AMC 12 Problems/Problem 1|Solution]]
  
 
== Problem 2 ==
 
== Problem 2 ==
 +
If <math>\sqrt {2 + \sqrt {x}} = 3</math>, then <math>x =</math>
 +
 +
A. 1
 +
B. <math>\sqrt {7}</math>
 +
C. 7
 +
D. 49
 +
E. 121
  
 
[[1995 AMC 12 Problems/Problem 2|Solution]]
 
[[1995 AMC 12 Problems/Problem 2|Solution]]
  
 
== Problem 3 ==
 
== Problem 3 ==
 +
The total in-store price for an appliance is <math>\</math>99.99<math>. A television commercial advertises the same product for three easy payments of </math>\<math>29.98</math> and a one-time shipping and handling charge of <math>\</math>9.98<math>. How much is saved by buying the appliance from the television advertiser?
 +
 +
A. 6 cents
 +
B. 7 cents
 +
C. 8 cents
 +
D. 9 cents
 +
E. 10 cents
  
 
[[1995 AMC 12 Problems/Problem 3|Solution]]
 
[[1995 AMC 12 Problems/Problem 3|Solution]]
  
 
== Problem 4 ==
 
== Problem 4 ==
 +
If </math>M<math> is </math>30 \%<math> of </math>Q<math>, </math>Q<math> is </math>20 \%<math> of </math>P<math>, and </math>N<math> is </math>50 \%<math> of </math>P<math>, then </math>\frac {M}{N} =<math>
 +
 +
A. </math>\frac {3}{250}<math>
 +
B. </math>\frac {3}{25}<math>
 +
C. 1
 +
D. </math>\displaystyle \frac {6}{5}<math>
 +
E. </math>\displaystyle \frac {4}{3}<math>
  
 
[[1995 AMC 12 Problems/Problem 4|Solution]]
 
[[1995 AMC 12 Problems/Problem 4|Solution]]
  
 
== Problem 5 ==
 
== Problem 5 ==
 +
A rectangular field is 300 feet wide and 400 feet long. Random sampling indicates that there are, on the average, three ants per square inch through out the field. [12 inches = 1 foot.] Of the following, the number that most closely approximates the number of ants in the field is
 +
 +
A. 500 thousand
 +
B. 5 million
 +
C. 50 million
 +
D. 500 million
 +
E. 5 billion
  
 
[[1995 AMC 12 Problems/Problem 5|Solution]]
 
[[1995 AMC 12 Problems/Problem 5|Solution]]
  
 
== Problem 6 ==
 
== Problem 6 ==
 +
The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked ?
 +
 +
{{image}}
 +
 +
A. A
 +
B. B
 +
C. C
 +
D. D
 +
E. E
 +
  
 
[[1995 AMC 12 Problems/Problem 6|Solution]]
 
[[1995 AMC 12 Problems/Problem 6|Solution]]
  
 
== Problem 7 ==
 
== Problem 7 ==
 +
The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a neglibile height above the equator, then, among the following choices, the best estimate of the number of hours of flight is:
 +
 +
A) 25
 +
B) 8
 +
C) 75
 +
D) 50
 +
E) 100
  
 
[[1995 AMC 12 Problems/Problem 7|Solution]]
 
[[1995 AMC 12 Problems/Problem 7|Solution]]
  
 
== Problem 8 ==
 
== Problem 8 ==
 +
In </math>\triangle ABC<math>, </math>\angle C = 90^\circ, AC = 6<math> and </math>BC = 8<math>. Points </math>D<math> and </math>E<math> are on </math>\overline{AB}<math> and </math>\overline{BC}<math>, respectively, and </math>\angle BED = 90^\circ<math>. If </math>DE = 4<math>, then </math>BD =<math>
 +
 +
A. 5
 +
B. </math>\displaystyle \frac {16}{3}<math>
 +
C. </math>\displaystyle \frac {20}{3}<math>
 +
D. </math>\displaystyle \frac {15}{2}<math>
 +
E. 8
  
 
[[1995 AMC 12 Problems/Problem 8|Solution]]
 
[[1995 AMC 12 Problems/Problem 8|Solution]]
  
 
== Problem 9 ==
 
== Problem 9 ==
 +
Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is
 +
 +
A. 10
 +
B. 12
 +
C. 14
 +
D. 16
 +
E. 18
  
 
[[1995 AMC 12 Problems/Problem 9|Solution]]
 
[[1995 AMC 12 Problems/Problem 9|Solution]]
  
 
== Problem 10 ==
 
== Problem 10 ==
 +
The area of the triangle bounded by the lines </math>y = x, y = - x<math> and </math>y = 6<math> is
 +
 +
A. 12
 +
B. </math>12 \sqrt 2<math>
 +
C. 24
 +
D. </math>24 \sqrt 2<math>
 +
E. 36
  
 
[[1995 AMC 12 Problems/Problem 10|Solution]]
 
[[1995 AMC 12 Problems/Problem 10|Solution]]
  
 
== Problem 11 ==
 
== Problem 11 ==
 +
How many base 10 four-digit numbers, </math>N = \underline{a} \underline{b} \underline{c} \underline{d}<math>, satisfy all three of the following conditions?
 +
 +
(i) </math>4,000 \leq N < 6,000;<math> (ii) </math>N<math> is a multiple of 5; (iii) </math>3 \leq b < c \leq 6<math>.
 +
 +
A. 10
 +
B. 18
 +
C. 24
 +
D. 36
 +
E. 48
  
 
[[1995 AMC 12 Problems/Problem 11|Solution]]
 
[[1995 AMC 12 Problems/Problem 11|Solution]]
  
 
== Problem 12 ==
 
== Problem 12 ==
 +
Let </math>f<math> be a linear function with the properties that </math>f(1) \leq f(2), f(3) \geq f(4),<math> and </math>f(5) = 5<math>. Which of the following is true?
 +
 +
A. </math>f(0) < 0<math>
 +
B. </math>f(0) = 0<math>
 +
C. </math>f(1) < f(0) < f( - 1)<math>
 +
D. </math>f(0) = 5<math>
 +
E. </math>f(0) > 5<math>
  
 
[[1995 AMC 12 Problems/Problem 12|Solution]]
 
[[1995 AMC 12 Problems/Problem 12|Solution]]
  
 
== Problem 13 ==
 
== Problem 13 ==
 +
The addition below is incorrect. The display can be made correct by changing one digit </math>d<math>, wherever it occurs, to another digit </math>e<math>. Find the sum of </math>d<math> and </math>e<math>.
 +
 +
{{image}}
 +
 +
Another table... Link: http://www.artofproblemsolving.com/Forum/viewtopic.php?p=373460#373460
 +
 +
A. 4
 +
B. 6
 +
C. 8
 +
D. 10
 +
E. More than 10
  
 
[[1995 AMC 12 Problems/Problem 13|Solution]]
 
[[1995 AMC 12 Problems/Problem 13|Solution]]
  
 
== Problem 14 ==
 
== Problem 14 ==
 +
If </math>f(x) = ax^4 - bx^2 + x + 5<math> and </math>f( - 3) = 2<math>, then </math>f(3) =<math>
 +
 +
A. -5
 +
B. -2
 +
C. 1
 +
D. 3
 +
E. 8
  
 
[[1995 AMC 12 Problems/Problem 14|Solution]]
 
[[1995 AMC 12 Problems/Problem 14|Solution]]
  
 
== Problem 15 ==
 
== Problem 15 ==
 +
Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps it will be on point
 +
 +
A. 1
 +
B. 2
 +
C. 3
 +
D. 4
 +
E. 5
  
 
[[1995 AMC 12 Problems/Problem 15|Solution]]
 
[[1995 AMC 12 Problems/Problem 15|Solution]]
  
 
== Problem 16 ==
 
== Problem 16 ==
 +
Anita attends a baseball game in Atlanta and estimates that there are 50,000 fans in attendance. Bob attends a baseball game in Boston and estimates that there are 60,000 fans in attendance. A league official who knows the actual numbers attending the two games note that:
 +
 +
i. The actual attendance in Atlanta is within </math>10 \%<math> of Anita's estimate.
 +
ii. Bob's estimate is within </math>10 \%<math> of the actual attendance in Boston.
 +
 +
To the nearest 1,000, the largest possible difference between the numbers attending the two games is
 +
 +
A. 10,000
 +
B. 11,000
 +
C. 20,000
 +
D. 21,000
 +
E. 22,000
  
 
[[1995 AMC 12 Problems/Problem 16|Solution]]
 
[[1995 AMC 12 Problems/Problem 16|Solution]]
  
 
== Problem 17 ==
 
== Problem 17 ==
 +
Given regular pentagon </math>ABCDE<math>, a circle can be drawn that is tangent to </math>\overline{DC}<math> at </math>D<math> and to </math>\overline{AB}<math> at </math>A<math>. The number of degrees in minor arc </math>AD<math> is
 +
 +
A. 72
 +
B. 108
 +
C. 120
 +
D. 135
 +
E. 144
 +
 +
{{image}}
  
 
[[1955 AMC 12 Problems/Problem 17|Solution]]
 
[[1955 AMC 12 Problems/Problem 17|Solution]]
  
 
== Problem 18 ==
 
== Problem 18 ==
 +
Two rays with common endpoint </math>O<math> forms a </math>30^\circ<math> angle. Point </math>A<math> lies on one ray, point </math>B<math> on the other ray, and </math>AB = 1<math>. The maximum possible length of </math>OB<math> is
 +
 +
A. 1
 +
B. </math>\displaystyle \frac {1 + \sqrt {3}}{\sqrt 2}<math>
 +
C. </math>\sqrt 3<math>
 +
D. 2
 +
E. </math>\frac {4}{\sqrt 3}<math>
  
 
[[1995 AMC 12 Problems/Problem 18|Solution]]
 
[[1995 AMC 12 Problems/Problem 18|Solution]]
  
 
== Problem 19 ==
 
== Problem 19 ==
 +
Equilateral triangle </math>DEF<math> is inscribed in equilateral triangle </math>ABC<math> such that </math>\overline{DE} \perp \overline{BC}<math>. The reatio of the area of </math>\triangle DEF<math> to the area of </math>\triangle ABC<math> is
 +
 +
A. </math>\displaystyle \frac {1}{6}<math>
 +
 +
B. </math>\displaystyle \frac {1}{4}<math>
 +
 +
C. </math>\displaystyle \frac {1}{3}<math>
 +
 +
D. </math>\displaystyle \frac {2}{5}<math>
 +
 +
E. </math>\displaystyle \frac {1}{2}<math>
  
 
[[1995 AMC 12 Problems/Problem 19|Solution]]
 
[[1995 AMC 12 Problems/Problem 19|Solution]]
  
 
== Problem 20 ==
 
== Problem 20 ==
−
If <math>a,b</math> and <math>c</math> are three (not necessarily different) numbers chosen randomly and with replacement from the set <math>\{1,2,3,4,5 \}</math>, the probability that <math>ab + c</math> is even is
+
If </math>a,b<math> and </math>c<math> are three (not necessarily different) numbers chosen randomly and with replacement from the set </math>\{1,2,3,4,5 \}<math>, the probability that </math>ab + c<math> is even is
  
  
−
<math> \mathrm{(A) \ \frac {2}{5} } \qquad \mathrm{(B) \ \frac {59}{125} } \qquad \mathrm{(C) \ \frac {1}{2} } \qquad \mathrm{(D) \ \frac {64}{125} } \qquad \mathrm{(E) \ \frac {3}{5} }  </math>
+
</math> \mathrm{(A) \ \frac {2}{5} } \qquad \mathrm{(B) \ \frac {59}{125} } \qquad \mathrm{(C) \ \frac {1}{2} } \qquad \mathrm{(D) \ \frac {64}{125} } \qquad \mathrm{(E) \ \frac {3}{5} }  <math>
  
 
[[1995 AMC 12 Problems/Problem 20|Solution]]
 
[[1995 AMC 12 Problems/Problem 20|Solution]]
  
 
== Problem 21 ==
 
== Problem 21 ==
 +
Two nonadjacent vertices of a rectangle are (4,3) and (-4,-3), and the coordinates of the other two vertices are integers. The number of such rectangles is
 +
 +
A. 1
 +
B. 2
 +
C. 3
 +
D. 4
 +
E. 5
  
 
[[1995 AMC 12 Problems/Problem 21|Solution]]
 
[[1995 AMC 12 Problems/Problem 21|Solution]]
Line 91: Line 251:
  
  
−
<math> \mathrm{(A) \ 459 } \qquad \mathrm{(B) \ 600 } \qquad \mathrm{(C) \ 680 } \qquad \mathrm{(D) \ 720 } \qquad \mathrm{(E) \ 745 }  </math>
+
</math> \mathrm{(A) \ 459 } \qquad \mathrm{(B) \ 600 } \qquad \mathrm{(C) \ 680 } \qquad \mathrm{(D) \ 720 } \qquad \mathrm{(E) \ 745 }  <math>
  
 
[[1995 AMC 12 Problems/Problem 22|Solution]]
 
[[1995 AMC 12 Problems/Problem 22|Solution]]
  
 
== Problem 23 ==
 
== Problem 23 ==
−
The sides of a triangle have lengths 11,15, and <math>k</math>, where <math>k</math> is an integer. For how many values of <math>k</math> is the triangle obtuse?
+
The sides of a triangle have lengths 11,15, and </math>k<math>, where </math>k<math> is an integer. For how many values of </math>k<math> is the triangle obtuse?
  
  
−
<math> \mathrm{(A) \ 5 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 12 } \qquad \mathrm{(D) \ 13 } \qquad \mathrm{(E) \ 14 }  </math>
+
</math> \mathrm{(A) \ 5 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 12 } \qquad \mathrm{(D) \ 13 } \qquad \mathrm{(E) \ 14 }  <math>
  
 
[[1995 AMC 12 Problems/Problem 23|Solution]]
 
[[1995 AMC 12 Problems/Problem 23|Solution]]
  
 
== Problem 24 ==
 
== Problem 24 ==
−
There exist positive integers <math>A,B</math> and <math>C</math>, with no common factor greater than 1, such that
+
There exist positive integers </math>A,B<math> and </math>C<math>, with no common factor greater than 1, such that
  
 
<cmath>A \log_{200} 5 + B \log_{200} 2 = C</cmath>
 
<cmath>A \log_{200} 5 + B \log_{200} 2 = C</cmath>
  
−
What is <math>A + B + C</math>?
+
What is </math>A + B + C<math>?
  
  
−
<math> \mathrm{(A) \ 6 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 9 } \qquad \mathrm{(E) \ 10 }  </math>
+
</math> \mathrm{(A) \ 6 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 9 } \qquad \mathrm{(E) \ 10 }  <math>
  
 
[[1995 AMC 12 Problems/Problem 24|Solution]]
 
[[1995 AMC 12 Problems/Problem 24|Solution]]
Line 119: Line 279:
  
  
−
<math> \mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ 12 }  </math>
+
</math> \mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ 12 }  <math>
  
 
[[1995 AMC 12 Problems/Problem 25|Solution]]
 
[[1995 AMC 12 Problems/Problem 25|Solution]]
  
 
== Problem 26 ==
 
== Problem 26 ==
−
In the figure, <math>\overline{AB}</math> and <math>\overline{CD}</math> are diameters of the circle with center <math>O</math>, <math>\overline{AB} \perp \overline{CD}</math>, and chord <math>\overline{DF}</math> intersects <math>\overline{AB}</math> at <math>E</math>. If <math>DE = 6</math> and <math>EF = 2</math>, then the area of the circle is  
+
In the figure, </math>\overline{AB}<math> and </math>\overline{CD}<math> are diameters of the circle with center </math>O<math>, </math>\overline{AB} \perp \overline{CD}<math>, and chord </math>\overline{DF}<math> intersects </math>\overline{AB}<math> at </math>E<math>. If </math>DE = 6<math> and </math>EF = 2<math>, then the area of the circle is  
  
  
Line 131: Line 291:
 
Link: http://www.artofproblemsolving.com/Forum/viewtopic.php?t=62477
 
Link: http://www.artofproblemsolving.com/Forum/viewtopic.php?t=62477
  
−
<math> \mathrm{(A) \ 23 \pi } \qquad \mathrm{(B) \ \frac {47}{2} \pi } \qquad \mathrm{(C) \ 24 \pi } \qquad \mathrm{(D) \ \frac {49}{2} \pi } \qquad \mathrm{(E) \ 25 \pi }  </math>
+
</math> \mathrm{(A) \ 23 \pi } \qquad \mathrm{(B) \ \frac {47}{2} \pi } \qquad \mathrm{(C) \ 24 \pi } \qquad \mathrm{(D) \ \frac {49}{2} \pi } \qquad \mathrm{(E) \ 25 \pi }  <math>
  
  
Line 143: Line 303:
 
A table really needs to go there... Link: http://www.artofproblemsolving.com/Forum/viewtopic.php?search_id=1526011879&t=62471
 
A table really needs to go there... Link: http://www.artofproblemsolving.com/Forum/viewtopic.php?search_id=1526011879&t=62471
  
−
Let <math>f(n)</math> denote the sum of the numbers in row <math>n</math>. What is the remainder when <math>f(100)</math> is divided by 100?
+
Let </math>f(n)<math> denote the sum of the numbers in row </math>n<math>. What is the remainder when </math>f(100)<math> is divided by 100?
  
  
−
<math> \mathrm{(A) \ 12 } \qquad \mathrm{(B) \ 30 } \qquad \mathrm{(C) \ 50 } \qquad \mathrm{(D) \ 62 } \qquad \mathrm{(E) \ 74 }  </math>
+
</math> \mathrm{(A) \ 12 } \qquad \mathrm{(B) \ 30 } \qquad \mathrm{(C) \ 50 } \qquad \mathrm{(D) \ 62 } \qquad \mathrm{(E) \ 74 }  <math>
  
 
[[1995 AMC 12 Problems/Problem 27|Solution]]
 
[[1995 AMC 12 Problems/Problem 27|Solution]]
  
 
== Problem 28 ==
 
== Problem 28 ==
−
Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of length <math>\sqrt {a}</math> where <math>a</math> is
+
Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of length </math>\sqrt {a}<math> where </math>a<math> is
  
  
−
<math> \mathrm{(A) \ 144 } \qquad \mathrm{(B) \ 156 } \qquad \mathrm{(C) \ 168 } \qquad \mathrm{(D) \ 176 } \qquad \mathrm{(E) \ 184 }  </math>
+
</math> \mathrm{(A) \ 144 } \qquad \mathrm{(B) \ 156 } \qquad \mathrm{(C) \ 168 } \qquad \mathrm{(D) \ 176 } \qquad \mathrm{(E) \ 184 }  <math>
  
 
[[1995 AMC 12 Problems/Problem 28|Solution]]
 
[[1995 AMC 12 Problems/Problem 28|Solution]]
  
 
== Problem 29 ==
 
== Problem 29 ==
−
For how many three-element sets of positive integers <math>\{a,b,c\}</math> is it true that <math>a \times b \times c = 2310</math>?
+
For how many three-element sets of positive integers </math>\{a,b,c\}<math> is it true that </math>a \times b \times c = 2310<math>?
  
  
−
<math> \mathrm{(A) \ 32 } \qquad \mathrm{(B) \ 36 } \qquad \mathrm{(C) \ 40 } \qquad \mathrm{(D) \ 43 } \qquad \mathrm{(E) \ 45 }  </math>
+
</math> \mathrm{(A) \ 32 } \qquad \mathrm{(B) \ 36 } \qquad \mathrm{(C) \ 40 } \qquad \mathrm{(D) \ 43 } \qquad \mathrm{(E) \ 45 }  <math>
  
 
[[1995 AMC 12 Problems/Problem 29|Solution]]
 
[[1995 AMC 12 Problems/Problem 29|Solution]]
Line 170: Line 330:
  
  
−
<math> \mathrm{(A) \ 16 } \qquad \mathrm{(B) \ 17 } \qquad \mathrm{(C) \ 18 } \qquad \mathrm{(D) \ 19 } \qquad \mathrm{(E) \ 20 }  </math>
+
</math> \mathrm{(A) \ 16 } \qquad \mathrm{(B) \ 17 } \qquad \mathrm{(C) \ 18 } \qquad \mathrm{(D) \ 19 } \qquad \mathrm{(E) \ 20 }  $
  
 
[[1995 AMC 12 Problems/Problem 30|Solution]]
 
[[1995 AMC 12 Problems/Problem 30|Solution]]

Revision as of 11:56, 8 January 2008

Problem 1

Kim earned scores of 87,83, and 88 on her first three mathematics examinations. If Kim receives a score of 90 on the fourth exam, then her average will

A. remain the same B. increase by 1 C. increase by 2 D. increase by 3 E. increase by 4

Solution

Problem 2

If $\sqrt {2 + \sqrt {x}} = 3$, then $x =$

A. 1 B. $\sqrt {7}$ C. 7 D. 49 E. 121

Solution

Problem 3

The total in-store price for an appliance is $$99.99$. A television commercial advertises the same product for three easy payments of$\$29.98$ and a one-time shipping and handling charge of $$9.98$. How much is saved by buying the appliance from the television advertiser?

A. 6 cents B. 7 cents C. 8 cents D. 9 cents E. 10 cents

[[1995 AMC 12 Problems/Problem 3|Solution]]

== Problem 4 == If$ (Error compiling LaTeX. Unknown error_msg)M$is$30 \%$of$Q$,$Q$is$20 \%$of$P$, and$N$is$50 \%$of$P$, then$\frac {M}{N} =$A.$\frac {3}{250}$B.$\frac {3}{25}$C. 1 D.$\displaystyle \frac {6}{5}$E.$\displaystyle \frac {4}{3}$[[1995 AMC 12 Problems/Problem 4|Solution]]

== Problem 5 == A rectangular field is 300 feet wide and 400 feet long. Random sampling indicates that there are, on the average, three ants per square inch through out the field. [12 inches = 1 foot.] Of the following, the number that most closely approximates the number of ants in the field is

A. 500 thousand B. 5 million C. 50 million D. 500 million E. 5 billion

[[1995 AMC 12 Problems/Problem 5|Solution]]

== Problem 6 == The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked ?

{{image}}

A. A B. B C. C D. D E. E


[[1995 AMC 12 Problems/Problem 6|Solution]]

== Problem 7 == The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a neglibile height above the equator, then, among the following choices, the best estimate of the number of hours of flight is:

A) 25 B) 8 C) 75 D) 50 E) 100

[[1995 AMC 12 Problems/Problem 7|Solution]]

== Problem 8 == In$ (Error compiling LaTeX. Unknown error_msg)\triangle ABC$,$\angle C = 90^\circ, AC = 6$and$BC = 8$. Points$D$and$E$are on$\overline{AB}$and$\overline{BC}$, respectively, and$\angle BED = 90^\circ$. If$DE = 4$, then$BD =$A. 5 B.$\displaystyle \frac {16}{3}$C.$\displaystyle \frac {20}{3}$D.$\displaystyle \frac {15}{2}$E. 8

[[1995 AMC 12 Problems/Problem 8|Solution]]

== Problem 9 == Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is

A. 10 B. 12 C. 14 D. 16 E. 18

[[1995 AMC 12 Problems/Problem 9|Solution]]

== Problem 10 == The area of the triangle bounded by the lines$ (Error compiling LaTeX. Unknown error_msg)y = x, y = - x$and$y = 6$is

A. 12 B.$ (Error compiling LaTeX. Unknown error_msg)12 \sqrt 2$C. 24 D.$24 \sqrt 2$E. 36

[[1995 AMC 12 Problems/Problem 10|Solution]]

== Problem 11 == How many base 10 four-digit numbers,$ (Error compiling LaTeX. Unknown error_msg)N = \underline{a} \underline{b} \underline{c} \underline{d}$, satisfy all three of the following conditions?

(i)$ (Error compiling LaTeX. Unknown error_msg)4,000 \leq N < 6,000;$(ii)$N$is a multiple of 5; (iii)$3 \leq b < c \leq 6$.

A. 10 B. 18 C. 24 D. 36 E. 48

[[1995 AMC 12 Problems/Problem 11|Solution]]

== Problem 12 == Let$ (Error compiling LaTeX. Unknown error_msg)f$be a linear function with the properties that$f(1) \leq f(2), f(3) \geq f(4),$and$f(5) = 5$. Which of the following is true?

A.$ (Error compiling LaTeX. Unknown error_msg)f(0) < 0$B.$f(0) = 0$C.$f(1) < f(0) < f( - 1)$D.$f(0) = 5$E.$f(0) > 5$[[1995 AMC 12 Problems/Problem 12|Solution]]

== Problem 13 == The addition below is incorrect. The display can be made correct by changing one digit$ (Error compiling LaTeX. Unknown error_msg)d$, wherever it occurs, to another digit$e$. Find the sum of$d$and$e$.

{{image}}

Another table... Link: http://www.artofproblemsolving.com/Forum/viewtopic.php?p=373460#373460

A. 4 B. 6 C. 8 D. 10 E. More than 10

[[1995 AMC 12 Problems/Problem 13|Solution]]

== Problem 14 == If$ (Error compiling LaTeX. Unknown error_msg)f(x) = ax^4 - bx^2 + x + 5$and$f( - 3) = 2$, then$f(3) =$A. -5 B. -2 C. 1 D. 3 E. 8

[[1995 AMC 12 Problems/Problem 14|Solution]]

== Problem 15 == Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps it will be on point

A. 1 B. 2 C. 3 D. 4 E. 5

[[1995 AMC 12 Problems/Problem 15|Solution]]

== Problem 16 == Anita attends a baseball game in Atlanta and estimates that there are 50,000 fans in attendance. Bob attends a baseball game in Boston and estimates that there are 60,000 fans in attendance. A league official who knows the actual numbers attending the two games note that:

i. The actual attendance in Atlanta is within$ (Error compiling LaTeX. Unknown error_msg)10 \%$of Anita's estimate. ii. Bob's estimate is within$10 \%$of the actual attendance in Boston.

To the nearest 1,000, the largest possible difference between the numbers attending the two games is

A. 10,000 B. 11,000 C. 20,000 D. 21,000 E. 22,000

[[1995 AMC 12 Problems/Problem 16|Solution]]

== Problem 17 == Given regular pentagon$ (Error compiling LaTeX. Unknown error_msg)ABCDE$, a circle can be drawn that is tangent to$\overline{DC}$at$D$and to$\overline{AB}$at$A$. The number of degrees in minor arc$AD$is

A. 72 B. 108 C. 120 D. 135 E. 144

{{image}}

[[1955 AMC 12 Problems/Problem 17|Solution]]

== Problem 18 == Two rays with common endpoint$ (Error compiling LaTeX. Unknown error_msg)O$forms a$30^\circ$angle. Point$A$lies on one ray, point$B$on the other ray, and$AB = 1$. The maximum possible length of$OB$is

A. 1 B.$ (Error compiling LaTeX. Unknown error_msg)\displaystyle \frac {1 + \sqrt {3}}{\sqrt 2}$C.$\sqrt 3$D. 2 E.$\frac {4}{\sqrt 3}$[[1995 AMC 12 Problems/Problem 18|Solution]]

== Problem 19 == Equilateral triangle$ (Error compiling LaTeX. Unknown error_msg)DEF$is inscribed in equilateral triangle$ABC$such that$\overline{DE} \perp \overline{BC}$. The reatio of the area of$\triangle DEF$to the area of$\triangle ABC$is

A.$ (Error compiling LaTeX. Unknown error_msg)\displaystyle \frac {1}{6}$B.$\displaystyle \frac {1}{4}$C.$\displaystyle \frac {1}{3}$D.$\displaystyle \frac {2}{5}$E.$\displaystyle \frac {1}{2}$[[1995 AMC 12 Problems/Problem 19|Solution]]

== Problem 20 == If$ (Error compiling LaTeX. Unknown error_msg)a,b$and$c$are three (not necessarily different) numbers chosen randomly and with replacement from the set$\{1,2,3,4,5 \}$, the probability that$ab + c$is even is$ \mathrm{(A) \ \frac {2}{5} } \qquad \mathrm{(B) \ \frac {59}{125} } \qquad \mathrm{(C) \ \frac {1}{2} } \qquad \mathrm{(D) \ \frac {64}{125} } \qquad \mathrm{(E) \ \frac {3}{5} } $[[1995 AMC 12 Problems/Problem 20|Solution]]

== Problem 21 == Two nonadjacent vertices of a rectangle are (4,3) and (-4,-3), and the coordinates of the other two vertices are integers. The number of such rectangles is

A. 1 B. 2 C. 3 D. 4 E. 5

[[1995 AMC 12 Problems/Problem 21|Solution]]

== Problem 22 == A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths 13,19,20,25 and 31, although this is not necessarily their order around the pentagon. The area of the pentagon is$ (Error compiling LaTeX. Unknown error_msg) \mathrm{(A) \ 459 } \qquad \mathrm{(B) \ 600 } \qquad \mathrm{(C) \ 680 } \qquad \mathrm{(D) \ 720 } \qquad \mathrm{(E) \ 745 } $[[1995 AMC 12 Problems/Problem 22|Solution]]

== Problem 23 == The sides of a triangle have lengths 11,15, and$ (Error compiling LaTeX. Unknown error_msg)k$, where$k$is an integer. For how many values of$k$is the triangle obtuse?$ \mathrm{(A) \ 5 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 12 } \qquad \mathrm{(D) \ 13 } \qquad \mathrm{(E) \ 14 } $[[1995 AMC 12 Problems/Problem 23|Solution]]

== Problem 24 == There exist positive integers$ (Error compiling LaTeX. Unknown error_msg)A,B$and$C$, with no common factor greater than 1, such that

<cmath>A \log_{200} 5 + B \log_{200} 2 = C</cmath>

What is$ (Error compiling LaTeX. Unknown error_msg)A + B + C$?$ \mathrm{(A) \ 6 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 9 } \qquad \mathrm{(E) \ 10 } $[[1995 AMC 12 Problems/Problem 24|Solution]]

== Problem 25 == A list of five positive integers has mean 12 and range 18. The mode and median are both 8. How many different values are possible for the second largest element of the list?$ (Error compiling LaTeX. Unknown error_msg) \mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ 12 } $[[1995 AMC 12 Problems/Problem 25|Solution]]

== Problem 26 == In the figure,$ (Error compiling LaTeX. Unknown error_msg)\overline{AB}$and$\overline{CD}$are diameters of the circle with center$O$,$\overline{AB} \perp \overline{CD}$, and chord$\overline{DF}$intersects$\overline{AB}$at$E$. If$DE = 6$and$EF = 2$, then the area of the circle is


{{image}}

Link: http://www.artofproblemsolving.com/Forum/viewtopic.php?t=62477$ (Error compiling LaTeX. Unknown error_msg) \mathrm{(A) \ 23 \pi } \qquad \mathrm{(B) \ \frac {47}{2} \pi } \qquad \mathrm{(C) \ 24 \pi } \qquad \mathrm{(D) \ \frac {49}{2} \pi } \qquad \mathrm{(E) \ 25 \pi } $[[1995 AMC 12 Problems/Problem 26|Solution]]

== Problem 27 == Consider the triangular array of numbers with 0,1,2,3,... along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown.

{{image}}

A table really needs to go there... Link: http://www.artofproblemsolving.com/Forum/viewtopic.php?search_id=1526011879&t=62471

Let$ (Error compiling LaTeX. Unknown error_msg)f(n)$denote the sum of the numbers in row$n$. What is the remainder when$f(100)$is divided by 100?$ \mathrm{(A) \ 12 } \qquad \mathrm{(B) \ 30 } \qquad \mathrm{(C) \ 50 } \qquad \mathrm{(D) \ 62 } \qquad \mathrm{(E) \ 74 } $[[1995 AMC 12 Problems/Problem 27|Solution]]

== Problem 28 == Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of length$ (Error compiling LaTeX. Unknown error_msg)\sqrt {a}$where$a$is$ \mathrm{(A) \ 144 } \qquad \mathrm{(B) \ 156 } \qquad \mathrm{(C) \ 168 } \qquad \mathrm{(D) \ 176 } \qquad \mathrm{(E) \ 184 } $[[1995 AMC 12 Problems/Problem 28|Solution]]

== Problem 29 == For how many three-element sets of positive integers$ (Error compiling LaTeX. Unknown error_msg)\{a,b,c\}$is it true that$a \times b \times c = 2310$?$ \mathrm{(A) \ 32 } \qquad \mathrm{(B) \ 36 } \qquad \mathrm{(C) \ 40 } \qquad \mathrm{(D) \ 43 } \qquad \mathrm{(E) \ 45 } $[[1995 AMC 12 Problems/Problem 29|Solution]]

== Problem 30 == A large cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is$ (Error compiling LaTeX. Unknown error_msg) \mathrm{(A) \ 16 } \qquad \mathrm{(B) \ 17 } \qquad \mathrm{(C) \ 18 } \qquad \mathrm{(D) \ 19 } \qquad \mathrm{(E) \ 20 } $

Solution

See also

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